Does the complex Langevin method give unbiased results
Does the complex Langevin method give unbiased results
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DOI：
10.1103/physrevd.94.114505
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发表时间：
2016-11
影响因子：
5
通讯作者：
L. L. Salcedo-L.
中科院分区：
文献类型：
--
作者：
L. L. Salcedo-L.
我们研究复杂 Langevin 算法的 Fokker-Planck 方程的平稳解是否再现了正确的期望值。当动作 $S(x)$ 的复数 Langevin 算法收敛时，它会产生等效的复数概率分布 $P(x)$，理想情况下与 $e^{-S(x)}$ 一致。我们证明由 $P(x)$ 满足的投影福克-普朗克方程可能包含一个其形式明确的异常项。该项破坏了关系 $P(x)=e^{-S(x)}$，在期望值中引入了偏差。通过对几个周期和非周期一维问题的分析，利用复平面上福克-普朗克方程的精确解或数值解，表明这种异常现象是普遍存在的。事实上，每当朗之万步行者只需要有限的时间去无穷大并返回时，就会出现异常，典型的动作就是这种情况。我们推测在一维情况下，异常是规则而不是例外，但是，随着涉及变量数量的增加，这种情况可能会发生变化。
We investigate whether the stationary solution of the Fokker-Planck equation of the complex Langevin algorithm reproduces the correct expectation values. When the complex Langevin algorithm for an action $S(x)$ is convergent, it produces an equivalent complex probability distribution $P(x)$ which ideally would coincide with $e^{-S(x)}$. We show that the projected Fokker-Planck equation fulfilled by $P(x)$ may contain an anomalous term whose form is made explicit. Such term spoils the relation $P(x)=e^{-S(x)}$, introducing a bias in the expectation values. Through the analysis of several periodic and non-periodic one-dimensional problems, using either exact or numerical solutions of the Fokker-Planck equation on the complex plane, it is shown that the anomaly is present quite generally. In fact, an anomaly is expected whenever the Langevin walker needs only a finite time to go to infinity and come back, and this is the case for typical actions. We conjecture that the anomaly is the rule rather than the exception in the one-dimensional case, however, this could change as the number of variables involved increases.